A Coloring Algorithm for Triangle-Free Graphs

We give a randomized algorithm that properly colors the vertices of a triangle-free graph G on n vertices using O(\Delta(G)/ log \Delta(G)) colors, where \Delta(G) is the maximum degree of G. The algorithm takes O(n\Delta2(G)log\Delta(G)) time and succeeds with high probability, provided \Delta(G) is greater than log^{1+{\epsilon}}n for a positive constant {\epsilon}. The number of colors is best possible up to a constant factor for triangle-free graphs. As a result this gives an algorithmic proof for a sharp upper bound of the chromatic number of a triangle-free graph, the existence of which was previously established by Kim and Johansson respectively.

[1]  Alexandr V. Kostochka,et al.  On an upper bound of a graph's chromatic number, depending on the graph's degree and density , 1977, J. Comb. Theory B.

[2]  Béla Bollobás.  Chromatic number, girth and maximal degree , 1978, Discret. Math..

[3]  Jeong Han Kim.  On Brooks' Theorem for Sparse Graphs , 1995, Comb. Probab. Comput..

[4]  Paul A. Catlin.  A bound on the chromatic number of a graph , 1978, Discret. Math..

[5]  János Komlós,et al.  A Dense Infinite Sidon Sequence , 1981, Eur. J. Comb..

[6]  N. Wormald.  Differential Equations for Random Processes and Random Graphs , 1995 .

[7]  Jim Lawrence.  Covering the vertex set of a graph with subgraphs of smaller degree , 1978, Discret. Math..

[8]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[9]  N. Alon,et al.  Wiley‐Interscience Series in Discrete Mathematics and Optimization , 2004 .

[10]  Subhash Khot,et al.  Improved inapproximability results for MaxClique, chromatic number and approximate graph coloring , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[11]  B. Reed.  Graph Colouring and the Probabilistic Method , 2001 .

[12]  Joel H. Spencer,et al.  Asymptotic behavior of the chromatic index for hypergraphs , 1989, J. Comb. Theory, Ser. A.

[13]  Vojtech Rödl,et al.  Near Perfect Coverings in Graphs and Hypergraphs , 1985, Eur. J. Comb..

[14]  A. V. Kostov cka,et al.  An estimate in the theory of graph coloring , 1977 .

[15]  V. G. Vizing.  SOME UNSOLVED PROBLEMS IN GRAPH THEORY , 1968 .

[16]  Jeong Han Kim,et al.  The Ramsey Number R(3, t) Has Order of Magnitude t2/log t , 1995, Random Struct. Algorithms.

[17]  R. L. Brooks.  On Colouring the Nodes of a Network , 1941 .

[18]  Alessandro Panconesi,et al.  Fast distributed algorithms for Brooks-Vizing colourings , 2000, SODA '98.

[19]  Van H. Vu,et al.  A General Upper Bound on the List Chromatic Number of Locally Sparse Graphs , 2002, Combinatorics, Probability and Computing.

[20]  Noga Alon,et al.  Coloring Graphs with Sparse Neighborhoods , 1999, J. Comb. Theory B.

[21]  Jeff Kahn,et al.  Coloring Nearly-Disjoint Hypergraphs with n+o(n) Colors , 1992, J. Comb. Theory, Ser. A.